Effective equidistribution in homogeneous dynamics, and local statistics of point sets of arithmetical origin

The proposed research centers on effective equidistribution in homogeneous dynamics and questions about the distribution of local spacings in various discrete sets of arithmetical origin.

I will focus in particular on problems which hold the promise of important applications in mathematical physics or in number theory.One part of the project is to prove effective versions of results on asymptotic equidistribution of unipotent orbits in specific settings within homogeneous dynamics - that is, to provide an explicit rate of for the equidistribution, in cases where previously only non-effective results are known.

For this task, I plan to use a blend of methods from harmonic analysis, analytic number theory and dynamics.Other parts of the project concern the fine-scale statistics of point sets of arithmetical origin.

For example, I wish to understand the asymptotic pair correlation density of the integer values of positive definite quadratic forms in 3 or more variables (subject to a suitable Diophantine condition); this problem is closely related to the Berry-Tabor conjecture on the quantum energy levels of a system whose classical dynamics is completely integrable.

Another problem under this heading concerns the fine-scale statistics of directions to the points of a quasicrystal; we will also investigate related questions regarding the Lorentz gas in a quasicrystalline scatterer configuration.